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Properties

Label 2-819-91.75-c0-0-0
Degree $2$
Conductor $819$
Sign $0.927 - 0.374i$
Analytic cond. $0.408734$
Root an. cond. $0.639323$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)13-s + 16-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (−1 − 1.73i)31-s − 1.73i·37-s + (−0.5 + 0.866i)43-s + (−0.499 − 0.866i)49-s + (0.5 + 0.866i)52-s + (−1.5 + 0.866i)61-s + 64-s + (0.5 + 0.866i)73-s + ⋯
L(s)  = 1  + 4-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)13-s + 16-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (−1 − 1.73i)31-s − 1.73i·37-s + (−0.5 + 0.866i)43-s + (−0.499 − 0.866i)49-s + (0.5 + 0.866i)52-s + (−1.5 + 0.866i)61-s + 64-s + (0.5 + 0.866i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(0.408734\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (712, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :0),\ 0.927 - 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.167539970\)
\(L(\frac12)\) \(\approx\) \(1.167539970\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 - T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.74505037510380759083610153538, −9.455047908595904740089513378444, −9.007512384843423488378465724023, −7.83286627745571123231207212917, −6.93205824052252072746307466285, −6.21266207958191977763269409951, −5.42237445505981380527847953511, −3.99473627762405494640583947296, −2.83030780568909639365597644881, −1.88083965047159551248193332931, 1.44493752625444309405013979992, 2.96742051011957795177639499258, 3.74998850517305259007644528119, 5.15023471642880211944818128004, 6.28381026013292702129446369852, 6.80844171650852591931944608237, 7.79448129832161713077602540064, 8.502498089016389182783005713106, 9.830152808512500670134016415317, 10.57980440693524486748472734183

Graph of the $Z$-function along the critical line